A375961 2-adic valuation of 6*n + 2.

1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 7, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 6, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 9, 1

Offset: 0

Author

Ruud H.G. van Tol, Sep 04 2024

Keywords

• nonn
• easy

Comments

6*i+2 is the first (3*x+1)/2 successor of 4*i+1, with i >= 0.

The first occurrence of odd t is before that of t-1.

Examples

			a(21) = A007814(6*21+2) = 7.
		

Links

• Ruud H.G. van Tol, Table of n, a(n) for n = 0..10000
• Index entries for sequences related to 3x+1 (or Collatz) problem.

Crossrefs

Cf. A007814, A016933, A087229, A087230, A096773, A371093, A375782.

Programs

Mathematica

a[n_] := IntegerExponent[6*n + 2, 2]; Array[a, 100, 0] (* Amiram Eldar, Sep 04 2024 *)

PARI

a(n) = valuation(6*n+2, 2);

Python

def A375961(n): return (~(3*n+1)&3*n).bit_length()+1 # Chai Wah Wu, Sep 27 2024

Formula

a(n) = A007814(6*n + 2).

a(n) = A371093(n) + 1.

a(n) = A087229(n) - 1.

a(n) = k for n == A096773(k) (mod 2^k), k >= 1.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2. - Amiram Eldar, Oct 01 2024